Theorem tsbi2 38824

Description: A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)

Assertion

Ref	Expression
tsbi2	⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓)))

Proof of Theorem tsbi2

Step	Hyp	Ref	Expression
1		pm5.21 837	. . . 4 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ 𝜓))
2	1	olcd 888	. . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓)))
3		pm4.57 1006	. . . . 5 ⊢ (¬ (¬ 𝜑 ∧ ¬ 𝜓) ↔ (𝜑 ∨ 𝜓))
4	3	biimpi 219	. . . 4 ⊢ (¬ (¬ 𝜑 ∧ ¬ 𝜓) → (𝜑 ∨ 𝜓))
5	4	orcd 887	. . 3 ⊢ (¬ (¬ 𝜑 ∧ ¬ 𝜓) → ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓)))
6	2, 5	pm2.61i 184	. 2 ⊢ ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓))
7	6	a1i 11	1 ⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862

This theorem is used by:  tsxo2  38828  mpobi123f  38852  mptbi12f  38856  ac6s6  38862